Here I would like to draw your attention to the idea of independence of two possibilities: whether an element belongs to a certain set or not. This idea occurred to me while trying to understand the nature of our dreams. Walking straight by the road you will face certain events and environment, while if you walk in the opposite direction, events and environment – everything will be quite different from what you have seen before. So, the two roads (straight and back) in your dreams are a kind of an asymmetric world represented to us as increased and dynamic.

On the one hand, if we try to analyze whether an element possibly belongs to a certain multitude we will apply specific notions. While on the other, if we try to prove that the element does not belong to this multitude we are likely to use other indexes.

Certainly, all the analyzed factors have much in common. However, this interrelation is rather complicated and implicit. (Let us suppose that they are multifractals, and that we need to establish the interrelation between their primings, which we do not know). The trick is that at a subconscious level we are able to connect, apparently untied factors, while at a conscious level we have not yet learned to do that.

For this case we have a dream machine. So, I have chosen the way of dream formation – having broken the connection between what can be with something that cannot be. Let me draw an example with dreams. Imagine, that today you have seen a beautiful building, which has seized your attention. Accordingly, it is highly possible that today in your night dream you will see some constructions. In the evening, watching news on TV, you see unpleasant staff of some international conflict. So, the possibility of seeing constructions in your dream still remains, while the possibility of not seeing them arises, as in your dream you may experience some kind of violence, etc. As a result, if moving straight in your dream you will come across some buildings, but try walking back and you will meet the war. Does not it remind you a compromise between our view of the world and our representation of the world?

Further on, trying to find the message of our dreams, I found out that we deal with interpretation of our problems in dreams in some aspects, which we had not noticed before when being awake. These aspects, however, are able to influence decision-making on certain problems. Is it worth moving straight if you will never get back. Then I tried to find linkage between my vision of the world in dreams and mathematics, and consequently between the notion of problem and some mathematical instrument.

It is not a secret that I am an admirer of mathematical machine of indistinct sets (fuzzy logic). Naturally, I either apply or plan to apply this instrument in all my projects. However, I often come across the problems of presence and absence of element in certain sets (actually the road back). Certainly you may assume that having calculated the probability of presence of element in the set I can easily estimate the probability of its absence in the set. Here that I supposed that these two probabilities may be absolutely different and have nothing in common. Moreover, the presence of element in the set and its absence get interrelated if we apply the same factors for analysis. While, actually it is just theory, and in practice factors influencing the presence or absence of element in a set are quite different.

Thus, the rarefied set can be described as follows: {e1(p1,z1),e2(p2,z2),e3(p3,z3)...en(pn,zn)} (the rarefied set), with “p” as probability of presence of “e” element in the set, and “z” as probability of its absence. I suppose, that the notion of «problem» can be well described with the help of such mathematical means. For this purpose let us analyze the following diagram.

The diagram shows the variations of uncertainty degree of analyzed sets according to the transition between the following notions: “problem — purpose — task”. One degree of probability disappears with each notion starting with the notion of problem! When analyzing problems we should consider the probabilities of fulfillment and non-fulfillment, while when analyzing purposes and tasks we should consider probability of achievement and a set of measures respectively.

The notion of risk is another idea, which requires some judgment. So I am not able to make any statements on this matter yet. However, I suppose that risk is somehow related to the probabilities of presence and absence of an element in a set.

The next step in development of idea should be the creation of some tool permitting to apply the idea in practice (something like Gant’s diagrams). The diagram includes the following marks: “M” as a set describing this or that notion, “e” as an element of a set (the factor under analysis), “p” as probability of presence of element in the set, “z” as probability of its absence in the set (here “p” and “z” are independent figures). In case with “PROBLEM” notion we deal with a rarefied set, where the two probabilities of presence and absence do not depend on each other. I think that mathematical means of complex figures could be applied here. However, it is just my supposition.

Let me draw another example fro the military field. Various factors influence probability of presence and absence of enemy forces in some specific place. For example, their presence is highly possible due to the tactic expedience. While, some climatic factors can influence the probability of their absence (World War II has shown that in Russia, for example, tanks are almost no use in rainy weather and in swampy areas). I mean that weather hardly depend on the battlefield situation.
Certainly, the idea on the rarefied sets still requires deep analysis from the point of view of the tool itself and its usage. However, its development and use in various fields of knowledge seems to be very expedient. It is enough to look at the diagram to see the variations of uncertainty degree of analyzed sets according to the transition between the following notions: “problem — purpose — task”.